Keir Starmer is not about to set a timetable for his departure from Downing Street, David Lammy, one of the prime minister’s closest cabinet allies, has said, urging Labour to get beyond the “spectacular own goal” of repeated leadership speculation.

The prime minister visited Labour HQ on Monday and told staff the whole party should show “100%” support to help win the crucial Makerfield byelection, as sources said he was not considering stepping aside for Andy Burnham should he win.

“There will be no timetable for departure,” Lammy, who is justice secretary and deputy prime minister, told Sky News. “Let me be really clear – Keir Starmer remains the most resilient person I know in my life. I spoke to him twice yesterday. He has a strength of character, a fighting experience. There will be no timetables.

“What there is, is getting on with the business of government, [he is] really crystal clear about that. At the moment, there is no contest. What there is, is his determination to deliver for the people.”

Burnham is expected to be selected for the Makerfield contest after the sitting Labour MP, Josh Simons, stepped down specifically so the Greater Manchester mayor could try to return to Westminster. If he wins against an expected strong challenger from Reform UK, he is likely to challenge for the leadership.

Speaking to Labour HQ staff, Starmer said the party must put the focus on what was being achieved in government. “It’s not been easy circumstances in the last 10 days. But you have just got on with the job that we asked you to do,” he said, going on to praise the GDP and growth figures as well as falls in NHS waiting times and the introduction of the Employment Rights Act and renters’ rights.

“The election results tell us that people are frustrated, they don’t feel that their lives have changed quickly enough,” he said. “We need to build up the urgency of what we do. We need a bit more hope in there. And we need to remember at all times what we are here to do. We were elected to government to serve the people of this country.”

Starmer did not mention Burnham by name because the mayor has not been officially selected but said everyone in the party should get out to defeat Reform.

“It is Labour versus Reform. We will know very shortly who the candidate is. Whoever they are I am going to support them 100% and I want every member, everyone in our movement to support them. A Labour candidate to beat Reform. That is the fight that we are in,” he said.

Lammy said Burnham would “be a great addition to parliament”, and that he would go to the constituency, on the edge of Wigan in Greater Manchester, to campaign for him.

But he lamented the week of internal wrangling, in which Simons stepped down and another potential challenger to Starmer, Wes Streeting, resigned as health secretary after seemingly failing to gather enough support from MPs to make a direct leadership challenge.

“I’m not going to sugar-coat this, I thought that the Labour party over the last 10 days had a spectacular own goal after those local election results,” Lammy said.

“We now need to unite and pull together. We have a byelection to fight. We need to remember our responsibilities as a government. We are not in opposition. We have the levers of power.”

Reform UK is expected to campaign heavily on Makerfield amid speculation that Labour might consider reversing Brexit, after Streeting used a speech on Saturday to say Britain’s long-term future lay in rejoining the union, something that was dismissed as “odd” by the culture secretary, Lisa Nandy.

Asked about the debate, Lammy said he was proud that, during his time as foreign secretary, he had rebuilt closer cooperation with EU member states, removed many trade barriers on food and agricultural products, and brought the UK back into the Erasmus student exchange scheme.

Asked about Streeting’s comments, Lammy said the government’s red lines – ruling out rejoining the EU’s customs union, single market or full membership – remained in place, adding: “I’m not going to make a commitment about the next election.”

Asked whether he would like the UK to rejoin one day, he said: “I’m committed to collective responsibility, the manifesto we stood on. Wes Streeting has left the government. He can have a debate, he can comment. That is not my position.”